The noble gases are helium, neon, argon, xenon and radon. The inertness of these elements is a consequence of the stability of the filled shells of electrons. The atomic numbers of the noble gases are 2, 10, 18, 36, 54 and 86. These can be considered magic numbers for electron structure stability. The differences in these numbers are: 8, 8, 18, 18, 32. These differences are twice the value of the squares of integers; i.e., 2(2²), 2(2²), 2(3²), 2(3²), 2(4²). The first number, 2, in the series {2, 10, 18, 36, 54, 86} is also of the form of twice the square of an integer, 2(1²).

The explanation of the magic numbers for electron structures is that there are shells for 2(n²) electrons where n=1, 2, 3, 4... The reason for the coefficient 2 in the formula is that there are two spin orientations of an electron. Pauli's exclusion principle operates and so electrons fill the states sequentially with no two electrons of an atom in the same state.

The reason for the squared integers is that the electrons have four quantum numbers (i, k, m, s). The first quantum number i can take on all integral values from −k to +k. That total (2k+1) possible states, an odd number. The quantum number k can take on all integral values from 0 to m. The sum of the first m odd numbers is m�. The quantum number s is the spin and s can take on only the values ��. Thus with the two spin orientation there are 2m� electrons with a quantum number m.

The Nuclear Magic Numbers

The conventional magic numbers for the nucleus are {2, 8, 20, 28, 50, 82, 126}. The case is made elsewhere that 6 and 14 are also magic numbers. (A recent study also found evidence for a magic number in the neighborhood of 180.) Thus the magic numbers for filled shells are then {2, 6, 8, 14, 20, 28, 50, 82, 126}.

Consider the following algorithm. Take the sequence of integers {0, 1, 2, 3, 4, 5, 6} and compute the cumulative sums; i.e., {0, 1, 3, 6, 10, 15, 21}. To each of these numbers add 1; i.e., {1, 2, 4, 7, 11, 16, 22}. Now double these numbers; i.e., {2, 4, 8, 14, 22, 32, 44}. Then compute the cumulative sums; i.e,,

{2, 6, 14, 28, 50, 82, 126}.

This is just the sequence of the nuclear magic numbers with 8 and 20 left out. There is no doubt that 8 and 20 are magic numbers. However in an analysis of the transition that takes place at magic number 28, the result which should depend upon the adjacent magic numbers is found to depend upon 14 and 50 rather than 20 and 50. It is perhaps possible that there are two classes of magic numbers and 8 and 20 are different from the other magic numbers. Also it is worth noting that 8 is the sum of the previous two magic numbers, 2 and 6, and 20 is likewise the sum of the two previous magic numbers in the above sequence, 6 and 14. It is worth noting that for electrons there are secondary peaks for the numbers 30, 48 and 80 and these numbers can be represented as sum of the electronic shell capacities. Thus for electronic shell structure there are primary magic numbers, {2, 10, 18, 36, 54, 86}, and secondary magic numbers {30, 48, 80}.

If the algorithm had been continued to 7 then the next magic number after 126 would be 184, clearly in the neighborhood of 180. If the algorithm is extended to 8, 9 and 10 the numbers generated are 258, 350 and 462. Some interesting things occur when a shell is half filled. That would be 304 for 258 and 350. The researchers that proposed 180 as the next magic number, more recently proposed 306 as the magic number to follow 180.

In the case of electrons, each electron in a shell is identified by a quadruplet of quantum numbers (i, k, m, s). For the nucleus, each neutron would be identified by a quadruplet of quantum numbers (p, q, n, s). The algorithm indicates that there would be some quantum number p taking on all integer values from 0 to q and q ranging from 0 to some value n. The doubling of the numbers corresponds to neutrons having two spin orientations, s=��. However the algorithm implies that there is for each shell some pair of states that is not identified by the above described set of quantum numbers.

For a justification for the above pattern of nuclear magic numbers see Nuclear Magic Numbers.

The Anomalous Singular State in Each Nucleonic Shell

The anomalous state is a problem and a puzzle but it may also be an opportunity to find out something about the structure of the shells. The states are filled sequentually, so the thing to look for is something interesting occurring for a particular point, such as the first, the middle, or the last state to be filled. Something interesting means a change in a pattern.

The incremental binding energy of a neutron for a nuclide is the difference between its binding energy and the binding energy of a nuclide having one less neutron. Hereafter incremental binding energy will refer to the incremental binding energy of a neutron. Consider the incremental binding energies for the element Indium (atomic number 49).

The sawtooth pattern is the effect of the formation of neutron-neutron spin pairs. The sharp drops after 50 and 82 neutrons are the result of the filling of neutron shells. Fifty and 82 are magic numbers. Those drops at 50 and 82 are interesting things but already explanable. The values at 51 and 52 do not fit into the pattern that prevails for 53 and above. Likewise the values for 80 and 81 do not fit into the pattern 53 through 79 neutrons. Something interesting occurs for the first and last neutrons in the shell.

Now consider the incremental binding energies of the isotopes of Stronium (atomic number 38).

The sharp drop after 38 neutrons occurs because that is the point where the number of neutrons equals the number of protons. There are 38 protons in the Stronium isotopes. Up to and including 38 each time a neutrons is added there is a neutron-proton spin pair formed. Beyond 38 there are no more such pairs formed. Thus the change in the pattern when the number of neutrons equals the number of protons is explanable.

In the above graph for the case of Strontium there is a perceptible change in the patter after 56 neutrons and then again after 59.

The case for the isotopes of Rubidium (atomic number 37) makes a good comparison with that of Stronium. After 37 the incremental binding energy is approximately unchanged because the increase due to the formation of a neutron-neutron spin pair almost exactly cancels out the decrease due to there not being a neutron-proton spin pair formed. It is of significance that the effects due to neutron-neutron spin pairs is essentially the same as that for neutron-proton spin pairs.

There appears to be a change in the pattern at 60 neutrons and perhaps at 56 neutrons, but definitely after 64 neutrons.

For Krypton, atomic number 36, there also is a change in the pattern after 56 neutrons.

For Bromine, atomic number 35, there is the now familiar near cancellation of the the change after 35 neutrons. There is a barely perceptible change after 56 neutrons. This is evidenced by the change in the slope of the pattern.

Although we are reaching the lower limits of elements having 56 neutrons it is worthwhile to look at two more cases.

For Selenium the continuation of the trend for the lower limits is different from the value for 57 neutrons. Likewise for Arsenic the continuation of the trend for the upper limits is different from the value for 56 neutrons. Thus for Selenium and Arsenic there is something interesting happening at 56 neutrons. Notably 56 is the sum of the magic numbers 50 and 6.

It is now worthwhile to consider the cases for larger atomic numbers to see if something interesting is happening for 56 neutrons. Yttrium has atomic number 39. The data for Yttrium are shown below.

Something interesting at 56 neutrons is occurring for Yttrium and likewise at 60 and 63 neutrons. Similar phenomena are displayed in the cases of Zirconium, and Niobium.

However the data for Molybdenum and elements with higher atomic numbers do not display these phenomena.

Conclusions

Nuclear magic numbers were originally defined in terms of the number of stable isotopes or isotones. There is a certain amount of ambiguity about the magic numbers defined in this way. The magicality of certain numbers can be defined unambiguously in terms of the incremental binding energies. At certain numbers the incremental binding energies for adding a neutron or adding a proton drops sharply. This test establishes the magicality of the traditional magic numbers {2, 8, 20, 28, 50, 82, 126} but also identifies 6 and 14 as magic numbers.

There is an algorithm based upon the existence of a set of four quantum numbers for each neutron or proton that generates the set of magic numbers {2, 6, 14, 28, 50, 82, 126}. This is the sequence of magic numbers with 8 and 20 left out.

The search for evidence of a special state in each nuclear shell did not reveal anything of that nature, but it did reveal that something special occurs for 56 neutrons and perhaps for 60 and 64 neutrons as well.

(To be continued.)